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An arithmetic sequence is a series of numbers in which each term after the first is formed by adding a constant to the preceding term. This constant is referred to as the "common difference." Consider the sequence: 1, 2, 3, ..., 10. Here, the common difference is 1 because you simply add 1 to each term to get the next.
This uniform increase enables the systematic approach of predicting further terms or summing the sequence using arithmetic methods. The formula for the nth term of an arithmetic sequence is given by:

• \(a_n = a_1 + (n-1)d\)

where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference. For this specific sequence, we see it starts at 1 and progresses by 1, which simplifies prediction of terms.

A mathematical expression is a combination of numbers, operations, and sometimes variables that stand for a particular value or set of values. It's like a sentence in the language of mathematics. For example, the sequence 1 + 2 + 3 + ... + 10 is a mathematical expression that represents a sum of consecutive integers.
Understanding mathematical expressions allows you to simplify, add, subtract, or even transform figures into different forms like sigma notation for better analysis or computation. Transforming an arithmetic sequence into sigma notation, as shown in the solution, is one way to succinctly express the sum of terms.

The general term is an expression that defines the position of a term within a sequence. It is crucial for summarizing the essence of the sequence without writing all individual terms. In an arithmetic sequence, each term can be expressed using the general formula:

• \( a_i = a_1 + (i-1) \,d \)

For the given sequence, since \( a_1 = 1 \) and the common difference \( d = 1 \), this simplifies to \( a_i = i \).
This simplification means merely substituting the number in the term's position (i.e., the index value). Through the general term, converting an entire list or sequence into sigma notation becomes clear and structured.